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19.05.2021, admin
Main menu Search. All about Infinity. Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. You may have met various ''paradoxes'' that play on this fact and our instinct to ignore it before. An infinitely long line, for instance, is surely infinitely many centimetres long.
It's also, equally surely, infinitely many miles long. But each centimetre is a great deal shorter than each mile, so does this mean an infinitely long line is two different lengths at once? The answer, of course, is to assert confidently that the question is meaningless and, if you're in the mood to be unkind, just shows how little the asker knows about proper mathematics and then go back about your business untroubled by such silly quibbles. Or, as I will hopefully convince you over the course of this article, the answer is a simple ''No.
We shouldn't treat infinity as a number. We can't count up to it, and if we have an infinitely large set we will never be able to count all of the objects in it. A set is a collection of objects. They can be any sort of objects, although generally mathematicians use them to talk about mathematical objects, such as numbers. An infinitely large set is, of course, one containing infinitely many objects.
It seems strange, therefore, to talk about certain infinitely large sets being ''countably infinite'', but that is indeed what I'm about to do. There are infinitely many natural numbers if there were only a finite number of them, there would have to be a largest natural number - what would happen if we added 1 to that number? We'll never be able to count them all. However, we can list them in such a way that if we counted forever, we'd be sure not to miss any out. The natural numbers are what we use for counting anyway, so we can think of them as coming ready-listed.
There is an obvious starting point 1 and a sensible order 1, 2, 3, To illustrate what I mean by a sensible order, imagine what would happen if we tried to count the natural numbers randomly.
However long we counted, we'd never be sure we'd counted them all - we could check for individual numbers in our random list, but we'd never know if they were all there, or when if ever we were going to reach a particular one.
Now imagine what would happen if we tried to count the natural numbers by counting the odd numbers first and then the even ones. We'd count forever, and never start counting the even numbers. However, if we count them in the order given, we'll know once we've reached that we've counted everything between 1 and once and only once.
We might never reach the end in a finite time, at least but we'll know we've not missed anything on the way and that it's just a matter of time before we reach any given number. That is, the only thing stopping us from counting them all is that we'll run out of breath before we run out of numbers. The proposed method of counting them all is sound, just impractical.
This is what we mean by saying the natural numbers are countably infinite. In much the same way, any infinite set of numbers that can be put in a sensible, systematic order with a clear beginning such that we're sure to get everything if we count forever is thought of as countable. If each cat does catch exactly one baby mouse - no more, no fewer - we know there are or, at least, were the same number of cats as mice.
This is called putting the objects into one-to-one correspondence, and the same can be done when comparing the sizes of infinite sets.
Another way of thinking about countably infinite sets is that they are those sets whose objects can be put in a one-to-one correspondence with the set containing the natural numbers only.
That is, if for each element in the set of the natural numbers there is exactly one - no more, no fewer - element in the set we're trying to count, then the set is the same size as the set containing the natural numbers: countably infinite. Putting Byjus Infinity Maths Quiz Characters the elements of an infinite set in sensible, systematic order with a clear beginning such that it's just a matter of time before we reach any particular element is, in fact, the same thing as putting those elements in one-to-one correspondence with the natural numbers.
If the natural numbers are the cats from the above illustration and the elements to be counted are the baby mice, then the cats already have an order to line up in. Lining the mice up next to them so none escape and none are eaten together putting them in a sensible order is the same as allotting the cats their dinner putting the elements of the two sets into one-to-one correspondence.
Are the integers all the whole numbers At first, you might think not. First we need to find a sensible beginning, and that may not be obvious. With the natural numbers, we just started at the smallest and worked up.
But the integers don't have a smallest number just as before, when showing there was no largest natural number, think to yourself: if there were a smallest integer, what would happen when we subtracted 1 from it? Say we start at 1 and work up, as before with the natural numbers. Then we count 1, 2, 3, Here, the problem is however long we count for, we'll never start on the negative numbers.
We'll never even get to zero. It is possible to do. Before reading on, try to think how. It may help to remember the problem we would have had counting the natural numbers if we had tried to count all the odd ones first.
It may help or it may not - don't dwell on this if it just confuses you to visualise the integers as distinct points on an infinitely long line. Think of 0 as the centre of the line, and imagine a circle, centre 0, which increases in diameter as we count up, covering the numbers we've counted.
After we've counted to the seventh integer on the list, say, the circle has a diameter of 6, and covers every number from -3 to 3. So, while at first glance it might seem that there are more integers than natural numbers, this is not the case. This is exactly what happens in the so-called paradox I mentioned at the start of this article. In the same way, you might think an infinite number of miles takes you further than an infinite number of centimetres.
In fact, the centimetres that go to make up the infinitely many miles can be put into one-to-one correspondence with the centimetres that go to make up the infinitely many centimeteres, so both take you the same distance.
How about the rational numbers? In the case of the natural numbers and the integers, it was easy to check we'd counted every number in a given range.
This is because between any two rationals, there is another rational. Between -2 and 4 there are seven integers including -2 and 4 and four natural numbers. There are, however, infinitely many rationals. How, then, could we possibly count them? We can never get anything in a given range, and there are infinitely many non-overlapping ranges we could be given.
In fact, if there are infinitely many naturals which there are and infinitely many integers which, again, there are then surely there must be infinity-squared many rationals, because to get all the rationals you take each integer in turn and divide it by each natural number in turn, as in the following table:. The first thing to do is to take a deep breath and remember that infinity is not a number.
In fact, the rationals are countable. To prove this, consider the above table. If it's on the table, then we know we've got all the positive rationals there. Is it there?
So, we have a complete list. However, trying to count along each row or column in turn gives problems - each row and column is infinitely long, after all, and so we'll never reach the end of the first to start on the second.
Is there, then, a way of counting them without missing any out? Just follow the red line in the image below:. So we know we've counted every rational at least once. Have we counted them at most once? Obviously, we haven't. Why does this matter? It's a good habit to get into, certainly, but more than that it's useful here to convince ourselves that these sets of numbers are all the same size.
If we don't check we're not counting the rationals too many times, what's to stop there being more natural numbers than there are rational ones? Intuitively, this seems to be a fairly silly fear after all, the natural numbers are just the first column of the entire table of rationals but if you're not doubting your intuition by this stage, I haven't explained what we're doing well enough.
We could just check each number we reach against all the previous numbers, making sure it's not equal to any of them. That would work, but ignores the fact that the order in which we're counting these numbers means the first time we meet each one, it's in its simplest possible form. Hence the positive rationals can be put into one-to-one correspondence with the naturals, and so are countably infinite.
Then, just as we did with the integers, start at 0 and interleave the two lists:. Thus the natural numbers are countable, the integers are countable and the rationals are countable. It seems as if everything is countable, and therefore all the infinite sets of numbers you can care to mention - even ones our intuition tells contain more objects than there are natural numbers - are the same size.
Are the real numbers countable? Every other set of numbers we've met so far has been countable. Each new set of numbers that feels as if it should be larger than the set of the natural numbers has been put into one-to-one correspondence with the natural numbers - all we needed was to work out how to list the numbers sensibly.
The real numbers are made up of the rationals and the irrationals. The rationals are countable, so if the irrationals are countable then the reals must be countable - just interleave our two systematic lists and we'll get another systematic list. For the same reason, if the reals aren't countable, we'll know that the problem comes with the irrationals. You've probably met rational numbers in at least two guises - one is that they can be written as one integer divided by another, and the other is that they can be written as decimal expansions that eventually become repeating patterns.
In fact, all real numbers can be represented as infinitely long decimal expansions. The rationals are the ones that eventually repeat and the irrationals are the ones that don't. Now, suppose the real numbers were countable. Then we could write a systematic list of all the real numbers. What is more, we could do it as list of decimal expansions. We might run into trouble with repeating a number without realising it 0.
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